Normalized defining polynomial

\( x^{22} - 2 x^{21} + 5 x^{20} + 14 x^{19} - 14 x^{18} + 142 x^{17} - 5 x^{16} + 272 x^{15} + 173 x^{14} + \cdots - 1 \) x^22 - 2*x^21 + 5*x^20 + 14*x^19 - 14*x^18 + 142*x^17 - 5*x^16 + 272*x^15 + 173*x^14 + 218*x^13 + 72*x^12 + 64*x^11 - 24*x^10 - 523*x^9 - 90*x^8 - 459*x^7 + 102*x^6 - 56*x^5 + 85*x^4 + 4*x^3 + 7*x^2 - 4*x - 1

Invariants

Degree:		$22$	sage: K.degree()   gp: poldegree(K.pol)   magma: Degree(K);   oscar: degree(K)
Signature:		$[6, 8]$	sage: K.signature()   gp: K.sign   magma: Signature(K);   oscar: signature(K)
Discriminant:		\(6168460699614248235671671923828125\) 6168460699614248235671671923828125 \(\medspace = 5^{11}\cdot 1831^{8}\)	sage: K.disc()   gp: K.disc   magma: OK := Integers(K); Discriminant(OK);   oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:		\(34.35\)	sage: (K.disc().abs())^(1./K.degree())   gp: abs(K.disc)^(1/poldegree(K.pol))   magma: Abs(Discriminant(OK))^(1/Degree(K));   oscar: (1.0 * dK)^(1/degree(K))
Galois root discriminant:		$5^{1/2}1831^{1/2}\approx 95.68176419778223$
Ramified primes:		\(5\), \(1831\) 5, 1831	sage: K.disc().support()   gp: factor(abs(K.disc))[,1]~   magma: PrimeDivisors(Discriminant(OK));   oscar: prime_divisors(discriminant((OK)))
Discriminant root field:		\(\Q(\sqrt{5}) \)
$\card{ \Aut(K/\Q) }$:		$2$	sage: K.automorphisms()   magma: Automorphisms(K);   oscar: automorphisms(K)
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{47}a^{20}+\frac{12}{47}a^{19}-\frac{7}{47}a^{18}+\frac{12}{47}a^{17}+\frac{4}{47}a^{16}+\frac{12}{47}a^{15}+\frac{7}{47}a^{14}-\frac{4}{47}a^{13}-\frac{15}{47}a^{12}+\frac{23}{47}a^{11}-\frac{8}{47}a^{10}-\frac{5}{47}a^{9}-\frac{17}{47}a^{8}-\frac{2}{47}a^{7}-\frac{19}{47}a^{6}+\frac{11}{47}a^{5}+\frac{10}{47}a^{4}-\frac{16}{47}a^{3}-\frac{12}{47}a^{2}-\frac{10}{47}a+\frac{6}{47}$, $\frac{1}{10\!\cdots\!67}a^{21}+\frac{38\!\cdots\!61}{10\!\cdots\!67}a^{20}-\frac{33\!\cdots\!53}{10\!\cdots\!67}a^{19}+\frac{33\!\cdots\!69}{10\!\cdots\!67}a^{18}+\frac{22\!\cdots\!31}{10\!\cdots\!67}a^{17}-\frac{38\!\cdots\!02}{10\!\cdots\!67}a^{16}+\frac{31\!\cdots\!34}{10\!\cdots\!67}a^{15}-\frac{36\!\cdots\!19}{10\!\cdots\!67}a^{14}+\frac{42\!\cdots\!03}{10\!\cdots\!67}a^{13}+\frac{15\!\cdots\!39}{10\!\cdots\!67}a^{12}-\frac{16\!\cdots\!87}{10\!\cdots\!67}a^{11}-\frac{54\!\cdots\!62}{10\!\cdots\!67}a^{10}+\frac{37\!\cdots\!33}{10\!\cdots\!67}a^{9}-\frac{35\!\cdots\!02}{10\!\cdots\!67}a^{8}+\frac{15\!\cdots\!28}{10\!\cdots\!67}a^{7}-\frac{13\!\cdots\!03}{10\!\cdots\!67}a^{6}+\frac{33\!\cdots\!48}{10\!\cdots\!67}a^{5}+\frac{10\!\cdots\!33}{10\!\cdots\!67}a^{4}+\frac{32\!\cdots\!59}{10\!\cdots\!67}a^{3}-\frac{29\!\cdots\!91}{10\!\cdots\!67}a^{2}+\frac{23\!\cdots\!34}{10\!\cdots\!67}a-\frac{44\!\cdots\!08}{10\!\cdots\!67}$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9, a^10, a^11, a^12, a^13, a^14, a^15, a^16, a^17, a^18, a^19, 1/47*a^20 + 12/47*a^19 - 7/47*a^18 + 12/47*a^17 + 4/47*a^16 + 12/47*a^15 + 7/47*a^14 - 4/47*a^13 - 15/47*a^12 + 23/47*a^11 - 8/47*a^10 - 5/47*a^9 - 17/47*a^8 - 2/47*a^7 - 19/47*a^6 + 11/47*a^5 + 10/47*a^4 - 16/47*a^3 - 12/47*a^2 - 10/47*a + 6/47, 1/101581987079819159741470267*a^21 + 383962489179151977598761/101581987079819159741470267*a^20 - 33023377970524802639988353/101581987079819159741470267*a^19 + 33889026890443552308413869/101581987079819159741470267*a^18 + 22654489864168760744877631/101581987079819159741470267*a^17 - 38185108264066552545843402/101581987079819159741470267*a^16 + 3165538680794952814968734/101581987079819159741470267*a^15 - 36041572542037962953121919/101581987079819159741470267*a^14 + 42576954608990110084476203/101581987079819159741470267*a^13 + 15710103842316891145999739/101581987079819159741470267*a^12 - 16289626381383339937472687/101581987079819159741470267*a^11 - 5496122526804980434940662/101581987079819159741470267*a^10 + 37162512315119568454840333/101581987079819159741470267*a^9 - 3540368379286097421618202/101581987079819159741470267*a^8 + 15311997519859427907881628/101581987079819159741470267*a^7 - 13068995704834252600688603/101581987079819159741470267*a^6 + 33104028772245654734585148/101581987079819159741470267*a^5 + 10542750634584345762230533/101581987079819159741470267*a^4 + 32552623596314552708376259/101581987079819159741470267*a^3 - 29323902608197723902278591/101581987079819159741470267*a^2 + 23482558155172561748135134/101581987079819159741470267*a - 44864396837923022862842708/101581987079819159741470267

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

Unit group

Rank:		$13$	sage: UK.rank()   gp: K.fu   magma: UnitRank(K);   oscar: rank(UK)
Torsion generator:		\( -1 \) -1  (order $2$)	sage: UK.torsion_generator()   gp: K.tu[2]   magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);   oscar: torsion_units_generator(OK)
Fundamental units:		$\frac{98\!\cdots\!00}{21\!\cdots\!61}a^{21}-\frac{15\!\cdots\!30}{21\!\cdots\!61}a^{20}+\frac{42\!\cdots\!22}{21\!\cdots\!61}a^{19}+\frac{15\!\cdots\!28}{21\!\cdots\!61}a^{18}-\frac{65\!\cdots\!38}{21\!\cdots\!61}a^{17}+\frac{13\!\cdots\!36}{21\!\cdots\!61}a^{16}+\frac{58\!\cdots\!32}{21\!\cdots\!61}a^{15}+\frac{29\!\cdots\!70}{21\!\cdots\!61}a^{14}+\frac{30\!\cdots\!39}{21\!\cdots\!61}a^{13}+\frac{35\!\cdots\!04}{21\!\cdots\!61}a^{12}+\frac{23\!\cdots\!90}{21\!\cdots\!61}a^{11}+\frac{17\!\cdots\!72}{21\!\cdots\!61}a^{10}+\frac{56\!\cdots\!38}{21\!\cdots\!61}a^{9}-\frac{48\!\cdots\!30}{21\!\cdots\!61}a^{8}-\frac{31\!\cdots\!79}{21\!\cdots\!61}a^{7}-\frac{59\!\cdots\!24}{21\!\cdots\!61}a^{6}-\frac{17\!\cdots\!72}{21\!\cdots\!61}a^{5}-\frac{13\!\cdots\!57}{21\!\cdots\!61}a^{4}+\frac{20\!\cdots\!80}{21\!\cdots\!61}a^{3}+\frac{13\!\cdots\!70}{21\!\cdots\!61}a^{2}+\frac{13\!\cdots\!98}{21\!\cdots\!61}a+\frac{56\!\cdots\!83}{21\!\cdots\!61}$, $\frac{22\!\cdots\!28}{10\!\cdots\!67}a^{21}-\frac{22\!\cdots\!22}{10\!\cdots\!67}a^{20}+\frac{75\!\cdots\!18}{10\!\cdots\!67}a^{19}+\frac{41\!\cdots\!99}{10\!\cdots\!67}a^{18}+\frac{52\!\cdots\!15}{10\!\cdots\!67}a^{17}+\frac{30\!\cdots\!94}{10\!\cdots\!67}a^{16}+\frac{30\!\cdots\!47}{10\!\cdots\!67}a^{15}+\frac{72\!\cdots\!98}{10\!\cdots\!67}a^{14}+\frac{10\!\cdots\!82}{10\!\cdots\!67}a^{13}+\frac{11\!\cdots\!95}{10\!\cdots\!67}a^{12}+\frac{78\!\cdots\!55}{10\!\cdots\!67}a^{11}+\frac{51\!\cdots\!66}{10\!\cdots\!67}a^{10}+\frac{19\!\cdots\!98}{10\!\cdots\!67}a^{9}-\frac{11\!\cdots\!01}{10\!\cdots\!67}a^{8}-\frac{14\!\cdots\!12}{10\!\cdots\!67}a^{7}-\frac{16\!\cdots\!96}{10\!\cdots\!67}a^{6}-\frac{80\!\cdots\!76}{10\!\cdots\!67}a^{5}-\frac{34\!\cdots\!91}{10\!\cdots\!67}a^{4}+\frac{11\!\cdots\!43}{10\!\cdots\!67}a^{3}+\frac{40\!\cdots\!96}{10\!\cdots\!67}a^{2}+\frac{59\!\cdots\!90}{10\!\cdots\!67}a+\frac{43\!\cdots\!16}{10\!\cdots\!67}$, $\frac{45\!\cdots\!70}{21\!\cdots\!61}a^{21}-\frac{70\!\cdots\!78}{21\!\cdots\!61}a^{20}+\frac{19\!\cdots\!28}{21\!\cdots\!61}a^{19}+\frac{72\!\cdots\!62}{21\!\cdots\!61}a^{18}-\frac{30\!\cdots\!64}{21\!\cdots\!61}a^{17}+\frac{63\!\cdots\!32}{21\!\cdots\!61}a^{16}+\frac{26\!\cdots\!70}{21\!\cdots\!61}a^{15}+\frac{13\!\cdots\!39}{21\!\cdots\!61}a^{14}+\frac{14\!\cdots\!04}{21\!\cdots\!61}a^{13}+\frac{16\!\cdots\!90}{21\!\cdots\!61}a^{12}+\frac{10\!\cdots\!72}{21\!\cdots\!61}a^{11}+\frac{80\!\cdots\!38}{21\!\cdots\!61}a^{10}+\frac{25\!\cdots\!70}{21\!\cdots\!61}a^{9}-\frac{22\!\cdots\!79}{21\!\cdots\!61}a^{8}-\frac{14\!\cdots\!24}{21\!\cdots\!61}a^{7}-\frac{27\!\cdots\!72}{21\!\cdots\!61}a^{6}-\frac{81\!\cdots\!57}{21\!\cdots\!61}a^{5}-\frac{63\!\cdots\!20}{21\!\cdots\!61}a^{4}+\frac{95\!\cdots\!70}{21\!\cdots\!61}a^{3}+\frac{62\!\cdots\!98}{21\!\cdots\!61}a^{2}+\frac{95\!\cdots\!83}{21\!\cdots\!61}a+\frac{98\!\cdots\!00}{21\!\cdots\!61}$, $\frac{13\!\cdots\!28}{10\!\cdots\!67}a^{21}-\frac{21\!\cdots\!57}{10\!\cdots\!67}a^{20}+\frac{57\!\cdots\!40}{10\!\cdots\!67}a^{19}+\frac{20\!\cdots\!85}{10\!\cdots\!67}a^{18}-\frac{10\!\cdots\!93}{10\!\cdots\!67}a^{17}+\frac{18\!\cdots\!74}{10\!\cdots\!67}a^{16}+\frac{66\!\cdots\!32}{10\!\cdots\!67}a^{15}+\frac{38\!\cdots\!73}{10\!\cdots\!67}a^{14}+\frac{38\!\cdots\!38}{10\!\cdots\!67}a^{13}+\frac{44\!\cdots\!36}{10\!\cdots\!67}a^{12}+\frac{26\!\cdots\!28}{10\!\cdots\!67}a^{11}+\frac{19\!\cdots\!66}{10\!\cdots\!67}a^{10}+\frac{47\!\cdots\!71}{10\!\cdots\!67}a^{9}-\frac{66\!\cdots\!91}{10\!\cdots\!67}a^{8}-\frac{38\!\cdots\!07}{10\!\cdots\!67}a^{7}-\frac{75\!\cdots\!15}{10\!\cdots\!67}a^{6}-\frac{16\!\cdots\!18}{10\!\cdots\!67}a^{5}-\frac{14\!\cdots\!13}{10\!\cdots\!67}a^{4}+\frac{51\!\cdots\!38}{10\!\cdots\!67}a^{3}+\frac{21\!\cdots\!46}{10\!\cdots\!67}a^{2}+\frac{16\!\cdots\!52}{10\!\cdots\!67}a+\frac{15\!\cdots\!34}{10\!\cdots\!67}$, $\frac{53\!\cdots\!63}{10\!\cdots\!67}a^{21}-\frac{79\!\cdots\!60}{10\!\cdots\!67}a^{20}+\frac{23\!\cdots\!52}{10\!\cdots\!67}a^{19}+\frac{84\!\cdots\!55}{10\!\cdots\!67}a^{18}-\frac{25\!\cdots\!61}{10\!\cdots\!67}a^{17}+\frac{75\!\cdots\!20}{10\!\cdots\!67}a^{16}+\frac{33\!\cdots\!71}{10\!\cdots\!67}a^{15}+\frac{17\!\cdots\!64}{10\!\cdots\!67}a^{14}+\frac{16\!\cdots\!12}{10\!\cdots\!67}a^{13}+\frac{22\!\cdots\!27}{10\!\cdots\!67}a^{12}+\frac{14\!\cdots\!65}{10\!\cdots\!67}a^{11}+\frac{10\!\cdots\!48}{10\!\cdots\!67}a^{10}+\frac{28\!\cdots\!67}{10\!\cdots\!67}a^{9}-\frac{26\!\cdots\!69}{10\!\cdots\!67}a^{8}-\frac{19\!\cdots\!88}{10\!\cdots\!67}a^{7}-\frac{37\!\cdots\!17}{10\!\cdots\!67}a^{6}-\frac{93\!\cdots\!52}{10\!\cdots\!67}a^{5}-\frac{92\!\cdots\!67}{10\!\cdots\!67}a^{4}+\frac{29\!\cdots\!27}{10\!\cdots\!67}a^{3}+\frac{78\!\cdots\!10}{10\!\cdots\!67}a^{2}+\frac{10\!\cdots\!16}{10\!\cdots\!67}a+\frac{54\!\cdots\!43}{10\!\cdots\!67}$, $\frac{15\!\cdots\!45}{10\!\cdots\!67}a^{21}-\frac{61\!\cdots\!29}{10\!\cdots\!67}a^{20}+\frac{12\!\cdots\!94}{10\!\cdots\!67}a^{19}-\frac{26\!\cdots\!92}{10\!\cdots\!67}a^{18}-\frac{85\!\cdots\!97}{10\!\cdots\!67}a^{17}+\frac{10\!\cdots\!88}{10\!\cdots\!67}a^{16}-\frac{81\!\cdots\!70}{10\!\cdots\!67}a^{15}+\frac{36\!\cdots\!30}{10\!\cdots\!67}a^{14}-\frac{14\!\cdots\!00}{10\!\cdots\!67}a^{13}-\frac{10\!\cdots\!75}{10\!\cdots\!67}a^{12}-\frac{11\!\cdots\!54}{10\!\cdots\!67}a^{11}-\frac{44\!\cdots\!74}{10\!\cdots\!67}a^{10}-\frac{37\!\cdots\!58}{10\!\cdots\!67}a^{9}-\frac{27\!\cdots\!46}{10\!\cdots\!67}a^{8}+\frac{30\!\cdots\!05}{10\!\cdots\!67}a^{7}+\frac{49\!\cdots\!25}{10\!\cdots\!67}a^{6}+\frac{24\!\cdots\!45}{10\!\cdots\!67}a^{5}-\frac{42\!\cdots\!45}{10\!\cdots\!67}a^{4}+\frac{22\!\cdots\!18}{10\!\cdots\!67}a^{3}-\frac{26\!\cdots\!10}{10\!\cdots\!67}a^{2}-\frac{40\!\cdots\!87}{10\!\cdots\!67}a+\frac{75\!\cdots\!39}{10\!\cdots\!67}$, $\frac{10\!\cdots\!54}{10\!\cdots\!67}a^{21}-\frac{15\!\cdots\!67}{10\!\cdots\!67}a^{20}+\frac{43\!\cdots\!65}{10\!\cdots\!67}a^{19}+\frac{16\!\cdots\!98}{10\!\cdots\!67}a^{18}-\frac{69\!\cdots\!27}{10\!\cdots\!67}a^{17}+\frac{14\!\cdots\!95}{10\!\cdots\!67}a^{16}+\frac{12\!\cdots\!89}{21\!\cdots\!61}a^{15}+\frac{29\!\cdots\!15}{10\!\cdots\!67}a^{14}+\frac{30\!\cdots\!37}{10\!\cdots\!67}a^{13}+\frac{35\!\cdots\!67}{10\!\cdots\!67}a^{12}+\frac{21\!\cdots\!28}{10\!\cdots\!67}a^{11}+\frac{14\!\cdots\!78}{10\!\cdots\!67}a^{10}+\frac{34\!\cdots\!85}{10\!\cdots\!67}a^{9}-\frac{52\!\cdots\!29}{10\!\cdots\!67}a^{8}-\frac{33\!\cdots\!50}{10\!\cdots\!67}a^{7}-\frac{59\!\cdots\!64}{10\!\cdots\!67}a^{6}-\frac{13\!\cdots\!07}{10\!\cdots\!67}a^{5}-\frac{10\!\cdots\!00}{10\!\cdots\!67}a^{4}+\frac{48\!\cdots\!80}{10\!\cdots\!67}a^{3}+\frac{17\!\cdots\!47}{10\!\cdots\!67}a^{2}+\frac{14\!\cdots\!46}{10\!\cdots\!67}a+\frac{67\!\cdots\!34}{10\!\cdots\!67}$, $\frac{40\!\cdots\!71}{10\!\cdots\!67}a^{21}-\frac{40\!\cdots\!59}{10\!\cdots\!67}a^{20}+\frac{14\!\cdots\!02}{10\!\cdots\!67}a^{19}+\frac{71\!\cdots\!73}{10\!\cdots\!67}a^{18}+\frac{12\!\cdots\!55}{10\!\cdots\!67}a^{17}+\frac{53\!\cdots\!26}{10\!\cdots\!67}a^{16}+\frac{50\!\cdots\!00}{10\!\cdots\!67}a^{15}+\frac{13\!\cdots\!69}{10\!\cdots\!67}a^{14}+\frac{16\!\cdots\!30}{10\!\cdots\!67}a^{13}+\frac{21\!\cdots\!79}{10\!\cdots\!67}a^{12}+\frac{14\!\cdots\!44}{10\!\cdots\!67}a^{11}+\frac{93\!\cdots\!98}{10\!\cdots\!67}a^{10}+\frac{34\!\cdots\!13}{10\!\cdots\!67}a^{9}-\frac{19\!\cdots\!33}{10\!\cdots\!67}a^{8}-\frac{24\!\cdots\!09}{10\!\cdots\!67}a^{7}-\frac{32\!\cdots\!16}{10\!\cdots\!67}a^{6}-\frac{12\!\cdots\!60}{10\!\cdots\!67}a^{5}-\frac{82\!\cdots\!03}{10\!\cdots\!67}a^{4}+\frac{25\!\cdots\!71}{10\!\cdots\!67}a^{3}+\frac{58\!\cdots\!90}{10\!\cdots\!67}a^{2}+\frac{11\!\cdots\!28}{10\!\cdots\!67}a+\frac{10\!\cdots\!28}{10\!\cdots\!67}$, $\frac{13\!\cdots\!59}{10\!\cdots\!67}a^{21}-\frac{26\!\cdots\!42}{10\!\cdots\!67}a^{20}+\frac{67\!\cdots\!53}{10\!\cdots\!67}a^{19}+\frac{18\!\cdots\!58}{10\!\cdots\!67}a^{18}-\frac{18\!\cdots\!21}{10\!\cdots\!67}a^{17}+\frac{18\!\cdots\!96}{10\!\cdots\!67}a^{16}-\frac{10\!\cdots\!50}{10\!\cdots\!67}a^{15}+\frac{37\!\cdots\!66}{10\!\cdots\!67}a^{14}+\frac{22\!\cdots\!55}{10\!\cdots\!67}a^{13}+\frac{31\!\cdots\!10}{10\!\cdots\!67}a^{12}+\frac{11\!\cdots\!13}{10\!\cdots\!67}a^{11}+\frac{11\!\cdots\!06}{10\!\cdots\!67}a^{10}-\frac{16\!\cdots\!23}{10\!\cdots\!67}a^{9}-\frac{67\!\cdots\!02}{10\!\cdots\!67}a^{8}-\frac{10\!\cdots\!04}{10\!\cdots\!67}a^{7}-\frac{66\!\cdots\!45}{10\!\cdots\!67}a^{6}+\frac{11\!\cdots\!80}{10\!\cdots\!67}a^{5}-\frac{13\!\cdots\!63}{10\!\cdots\!67}a^{4}+\frac{10\!\cdots\!66}{10\!\cdots\!67}a^{3}-\frac{54\!\cdots\!02}{10\!\cdots\!67}a^{2}+\frac{14\!\cdots\!93}{10\!\cdots\!67}a-\frac{45\!\cdots\!19}{10\!\cdots\!67}$, $\frac{40\!\cdots\!19}{10\!\cdots\!67}a^{21}-\frac{69\!\cdots\!91}{10\!\cdots\!67}a^{20}+\frac{16\!\cdots\!68}{10\!\cdots\!67}a^{19}+\frac{66\!\cdots\!36}{10\!\cdots\!67}a^{18}-\frac{47\!\cdots\!23}{10\!\cdots\!67}a^{17}+\frac{54\!\cdots\!51}{10\!\cdots\!67}a^{16}+\frac{18\!\cdots\!33}{10\!\cdots\!67}a^{15}+\frac{92\!\cdots\!29}{10\!\cdots\!67}a^{14}+\frac{11\!\cdots\!52}{10\!\cdots\!67}a^{13}+\frac{87\!\cdots\!96}{10\!\cdots\!67}a^{12}+\frac{59\!\cdots\!14}{10\!\cdots\!67}a^{11}+\frac{46\!\cdots\!94}{10\!\cdots\!67}a^{10}+\frac{21\!\cdots\!68}{10\!\cdots\!67}a^{9}-\frac{20\!\cdots\!83}{10\!\cdots\!67}a^{8}-\frac{88\!\cdots\!43}{10\!\cdots\!67}a^{7}-\frac{13\!\cdots\!22}{10\!\cdots\!67}a^{6}-\frac{62\!\cdots\!38}{10\!\cdots\!67}a^{5}-\frac{42\!\cdots\!79}{10\!\cdots\!67}a^{4}-\frac{22\!\cdots\!60}{10\!\cdots\!67}a^{3}+\frac{53\!\cdots\!74}{10\!\cdots\!67}a^{2}+\frac{27\!\cdots\!68}{10\!\cdots\!67}a+\frac{31\!\cdots\!33}{10\!\cdots\!67}$, $\frac{21\!\cdots\!49}{21\!\cdots\!61}a^{21}-\frac{18\!\cdots\!82}{10\!\cdots\!67}a^{20}+\frac{46\!\cdots\!37}{10\!\cdots\!67}a^{19}+\frac{15\!\cdots\!57}{10\!\cdots\!67}a^{18}-\frac{12\!\cdots\!90}{10\!\cdots\!67}a^{17}+\frac{13\!\cdots\!55}{10\!\cdots\!67}a^{16}+\frac{19\!\cdots\!76}{10\!\cdots\!67}a^{15}+\frac{26\!\cdots\!27}{10\!\cdots\!67}a^{14}+\frac{22\!\cdots\!06}{10\!\cdots\!67}a^{13}+\frac{23\!\cdots\!82}{10\!\cdots\!67}a^{12}+\frac{11\!\cdots\!24}{10\!\cdots\!67}a^{11}+\frac{94\!\cdots\!65}{10\!\cdots\!67}a^{10}+\frac{10\!\cdots\!60}{10\!\cdots\!67}a^{9}-\frac{51\!\cdots\!49}{10\!\cdots\!67}a^{8}-\frac{16\!\cdots\!74}{10\!\cdots\!67}a^{7}-\frac{43\!\cdots\!94}{10\!\cdots\!67}a^{6}-\frac{14\!\cdots\!00}{10\!\cdots\!67}a^{5}-\frac{57\!\cdots\!96}{10\!\cdots\!67}a^{4}+\frac{35\!\cdots\!49}{10\!\cdots\!67}a^{3}+\frac{69\!\cdots\!78}{10\!\cdots\!67}a^{2}+\frac{76\!\cdots\!61}{10\!\cdots\!67}a+\frac{99\!\cdots\!26}{10\!\cdots\!67}$, $\frac{17\!\cdots\!00}{10\!\cdots\!67}a^{21}-\frac{50\!\cdots\!34}{10\!\cdots\!67}a^{20}+\frac{11\!\cdots\!81}{10\!\cdots\!67}a^{19}+\frac{19\!\cdots\!25}{10\!\cdots\!67}a^{18}-\frac{50\!\cdots\!50}{10\!\cdots\!67}a^{17}+\frac{26\!\cdots\!19}{10\!\cdots\!67}a^{16}-\frac{20\!\cdots\!31}{10\!\cdots\!67}a^{15}+\frac{40\!\cdots\!06}{10\!\cdots\!67}a^{14}-\frac{12\!\cdots\!68}{10\!\cdots\!67}a^{13}-\frac{11\!\cdots\!79}{10\!\cdots\!67}a^{12}-\frac{16\!\cdots\!19}{10\!\cdots\!67}a^{11}+\frac{25\!\cdots\!32}{10\!\cdots\!67}a^{10}-\frac{75\!\cdots\!46}{10\!\cdots\!67}a^{9}-\frac{90\!\cdots\!10}{10\!\cdots\!67}a^{8}+\frac{65\!\cdots\!83}{10\!\cdots\!67}a^{7}-\frac{39\!\cdots\!03}{10\!\cdots\!67}a^{6}+\frac{57\!\cdots\!18}{10\!\cdots\!67}a^{5}-\frac{11\!\cdots\!69}{10\!\cdots\!67}a^{4}+\frac{11\!\cdots\!07}{10\!\cdots\!67}a^{3}-\frac{49\!\cdots\!11}{10\!\cdots\!67}a^{2}+\frac{54\!\cdots\!57}{10\!\cdots\!67}a+\frac{35\!\cdots\!39}{10\!\cdots\!67}$, $\frac{36\!\cdots\!78}{10\!\cdots\!67}a^{21}-\frac{58\!\cdots\!14}{10\!\cdots\!67}a^{20}+\frac{16\!\cdots\!45}{10\!\cdots\!67}a^{19}+\frac{58\!\cdots\!93}{10\!\cdots\!67}a^{18}-\frac{26\!\cdots\!03}{10\!\cdots\!67}a^{17}+\frac{51\!\cdots\!83}{10\!\cdots\!67}a^{16}+\frac{19\!\cdots\!73}{10\!\cdots\!67}a^{15}+\frac{10\!\cdots\!38}{10\!\cdots\!67}a^{14}+\frac{10\!\cdots\!44}{10\!\cdots\!67}a^{13}+\frac{12\!\cdots\!90}{10\!\cdots\!67}a^{12}+\frac{80\!\cdots\!60}{10\!\cdots\!67}a^{11}+\frac{57\!\cdots\!42}{10\!\cdots\!67}a^{10}+\frac{15\!\cdots\!07}{10\!\cdots\!67}a^{9}-\frac{18\!\cdots\!27}{10\!\cdots\!67}a^{8}-\frac{11\!\cdots\!13}{10\!\cdots\!67}a^{7}-\frac{21\!\cdots\!04}{10\!\cdots\!67}a^{6}-\frac{53\!\cdots\!31}{10\!\cdots\!67}a^{5}-\frac{45\!\cdots\!71}{10\!\cdots\!67}a^{4}+\frac{14\!\cdots\!49}{10\!\cdots\!67}a^{3}+\frac{69\!\cdots\!04}{10\!\cdots\!67}a^{2}+\frac{59\!\cdots\!77}{10\!\cdots\!67}a+\frac{97\!\cdots\!32}{10\!\cdots\!67}$ 985861346188691665992900/2161318874038705526414261*a^21 - 1516187051703137164494030/2161318874038705526414261*a^20 + 4227553937101641295633122/2161318874038705526414261*a^19 + 15758316372541608987979228/2161318874038705526414261*a^18 - 6526743810919434344956338/2161318874038705526414261*a^17 + 136963547017363050532793936/2161318874038705526414261*a^16 + 58378242057717539554123532/2161318874038705526414261*a^15 + 294978059282705703201525970/2161318874038705526414261*a^14 + 306950125710197613247789539/2161318874038705526414261*a^13 + 356480293492726355720954004/2161318874038705526414261*a^12 + 235930885609303681217704690/2161318874038705526414261*a^11 + 171966957725983550164873672/2161318874038705526414261*a^10 + 56360639807107224954586738/2161318874038705526414261*a^9 - 489780360287138114080207730/2161318874038705526414261*a^8 - 314771559604632349167429179/2161318874038705526414261*a^7 - 598270704468424269874296824/2161318874038705526414261*a^6 - 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1170920491872993468105648807/101581987079819159741470267*a^3 - 4907147233702319708175360411/101581987079819159741470267*a^2 + 542949558341475076954725757/101581987079819159741470267*a + 357295905395481748867475639/101581987079819159741470267, 369213052361912137272514078/101581987079819159741470267*a^21 - 583345466440809220686069814/101581987079819159741470267*a^20 + 1607032619517498526797408245/101581987079819159741470267*a^19 + 5828159801624540422290682093/101581987079819159741470267*a^18 - 2683222518160008840365703703/101581987079819159741470267*a^17 + 51366144648599789481865655683/101581987079819159741470267*a^16 + 19590407783518186165798667073/101581987079819159741470267*a^15 + 109564859652131039789109216438/101581987079819159741470267*a^14 + 109319998607433168513357319644/101581987079819159741470267*a^13 + 128055055577274318362775951990/101581987079819159741470267*a^12 + 80448272076506855843966130560/101581987079819159741470267*a^11 + 57982392583383579314393901342/101581987079819159741470267*a^10 + 15286620849293430969003005807/101581987079819159741470267*a^9 - 186663573042627870541580612527/101581987079819159741470267*a^8 - 112027934940370902515749298513/101581987079819159741470267*a^7 - 219636248828055641642026148704/101581987079819159741470267*a^6 - 53297957628155676664241827131/101581987079819159741470267*a^5 - 45633012842970955941487744971/101581987079819159741470267*a^4 + 14255082616848185083488274349/101581987079819159741470267*a^3 + 6923644416814126473834363504/101581987079819159741470267*a^2 + 5978220261705767524166214077/101581987079819159741470267*a + 979650425256050409279892232/101581987079819159741470267 (assuming GRH)	sage: UK.fundamental_units()   gp: K.fu   magma: [K|fUK(g): g in Generators(UK)];   oscar: [K(fUK(a)) for a in gens(UK)]
Regulator:		\( 185058540.947 \) (assuming GRH)	sage: K.regulator()   gp: K.reg   magma: Regulator(K);   oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{8}\cdot 185058540.947 \cdot 1}{2\cdot\sqrt{6168460699614248235671671923828125}}\cr\approx \mathstrut & 0.183151153351 \end{aligned}\] (assuming GRH)

Galois group

$C_2\times \PSL(2,11)$ (as 22T13):

Intermediate fields

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Frobenius cycle types

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

Local algebras for ramified primes

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
\(5\) 5	5.22.11.1	$x^{22} + 220 x^{21} + 22055 x^{20} + 1331000 x^{19} + 53791375 x^{18} + 1531447500 x^{17} + 31435820625 x^{16} + 467679300000 x^{15} + 4991151206250 x^{14} + 37171668875000 x^{13} + 183624733943756 x^{12} + 553513923250726 x^{11} + 918123669784090 x^{10} + 929291725767350 x^{9} + 623894056087500 x^{8} + 292303912609500 x^{7} + 98324330218125 x^{6} + 25190924781000 x^{5} + 17099014728125 x^{4} + 90189081743750 x^{3} + 391939091809384 x^{2} + 906877245981448 x + 669277565422109$	$2$	$11$	$11$	22T1	$[\ ]_{2}^{11}$
\(1831\) 1831		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $3$	$1$	$3$	$0$	$C_3$	$[\ ]^{3}$
		Deg $3$	$1$	$3$	$0$	$C_3$	$[\ ]^{3}$
		Deg $6$	$2$	$3$	$3$
		Deg $6$	$2$	$3$	$3$